Applications of an operator that is never assembled, against the entries of the matrix it stands for
At its defaults it draws applications of an operator that is never assembled, against the entries of the matrix it stands for. The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 112, 160, 208, 256 at n = 64, 128, 256, 512 — 48 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 256 products against 262,144 entries, and the representation it produces is within 7.3× of the one that read them all.
products-count is one function in lib/figures/hbuild.js —
built from products — a few hundred applications of an operator that is never formed. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 112, 160, 208, 256 at n = 64, 128, 256, 512 — 48 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 256 products against 262,144 entries, and the representation it produces is within 7.3× of the one that read them all.
k: 10
The arguments are the ones Built from products alone passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 112, 160, 208, 256 at n = 64, 128, 256, 512 — 48 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 256 products against 262,144 entries, and the representation it produces is within 7.3× of the one that read them all.
k: 4
The arguments are the ones Built from products alone passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 64, 88, 112, 136 at n = 64, 128, 256, 512 — 24 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 136 products against 262,144 entries, and the representation it produces is within 10.7× of the one that read them all.
k: 6
The arguments are the ones Built from products alone passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 80, 112, 144, 176 at n = 64, 128, 256, 512 — 32 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 176 products against 262,144 entries, and the representation it produces is within 8.0× of the one that read them all.
k: 8
The arguments are the ones Built from products alone passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 96, 136, 176, 216 at n = 64, 128, 256, 512 — 40 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 216 products against 262,144 entries, and the representation it produces is within 7.1× of the one that read them all.
k: 14
The arguments are the ones Built from products alone passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The blocks at one level of the tree have disjoint column supports, so one batch of random vectors samples all of them at once: two batches a level for the ranges, two more for the projections, and one batch of leaf products for every diagonal block in the matrix together. That is leaf + 2(2k + p)·levels products, and the counter inside the operator says 144, 208, 272, 336 at n = 64, 128, 256, 512 — 64 more per doubling, which is a logarithm. The other line is n², the entries the compression route reads. At n = 512 that is 336 products against 262,144 entries, and the representation it produces is within 6.5× of the one that read them all.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
10 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
and the representation is within a factor of the best of its own rank at n = 64 — checked 4 times
a rank the leaf blocks can hold
a rank the leaf blocks can hold and the sweep resolves
an oversampling inside the range this is worth drawing over
matmul shapes agree
so products per unknown falls at every doubling
the number of products grows by a constant per doubling, which is a logarithm
Against the rule
It draws a decomposition and prints its residual. It calls
productsAgainstSize,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Built from products alone
A 512-square hierarchical representation, at a relative error of 4·10⁻⁷, from 256 applications of an operator that is never assembled. The compression route reads 262,144 entries; this one reads none, and pays for it with a factor of seven against the representation the entries would have given.
Where the flop count stopped predicting the timeWhere the format starts paying
A hierarchical solve costs 1.48 times a dense factorisation at 64 unknowns and 0.16 times it at 512. The crossover is between 64 and 128, it walks right when the accuracy is tightened, and the exponent between consecutive sizes is 2.13, 1.93, 1.74 — falling towards one and never arriving.